How to find the square root of a non-perfect square manually

Although it is really really simple to get the answer for "Square root of 598" using a calculator within milliseconds. However, how to get the answer or the decimal value manually might be problematic. If there is a school homework or need to try it before the exam, it is quite easy and few little steps to keep in mind. Here we go with the required manual process to find out the square root of a non-perfect square!

There are only three steps to do!

1. Take the closest minimum perfect square to the number (here, the number is 598) and take the square root of it. Obviously, it MUST BE a positive integer!
2. Then, divide the number by the taken square root (or by the average value from step 3)
3. Get the average of the square root and result of the step 2

That is it, you have got the way !

However, there are several things to think about. The answers might depend on the number of decimal places required. It is clear that answers must be infinite decimals due to use of irrational numbers. Therefore, the answers must be compacted into some length as required, otherwise the whole process can not be stopped ever. That is the power of infinite decimals. (If you need more about irrational numbers and infinite decimals, you can learn them from link 1 and link 2 respectively!)

To get the most accurate answers, it should be done in the following manner. Think about the above three (1, 2, and 3) steps as one process.

1. Need only one decimal place :- Do the process just once and round the answer to one decimal place.
2. Need two decimal places :- Do the process once and repeat step 2 and 3 for another time. Then, round the answer to two decimal places.
3. Need three decimal places :- Do the process once and repeat step 2 and 3 for more two times. Then, round the answer to three decimal places.

An example can be used to clearly show this in a simple way as below.

Let's take, 

 
It is required to find the square root in one decimal place, in two decimal places, and in three decimal places.

Since 576 is the closest minimum perfect square for 598, the answer to step one in the process is 24.

Figure 1 shows the calculation to find for one decimal place below.

Figure 1

Figure 2 shows the calculation to find two decimal places below.

Figure 2

Figure 3 shows the calculation to find three decimal places below.

Figure 3

IMPORTANT; When getting the answers with more than one decimal places, calculations should be done corresponding times as mentioned above. Otherwise the answers are not accurate. Let’s discuss the situation considering the above example using the answers in figure 1, 2, and 3.

This is from figure 1, after rounding the average value into first decimal place.

However, the average value is approximately 24.458333 from a single iteration of the process. Therefore, if we choose 24.45 and 24.458 as answers for second and third cases respectively, it is a blind wrong! It can be shown by comparing the correct answers taken from figures and thought values here. For a single decimal place, it is totally fine from one iteration! Figure 4 shows the explanation for two decimal places and figure 5 shows the explanation for three decimal places as below.

Figure 4

Figure 5

Finally, it is not a difficult procedure to follow and you just need to calculate and getting values very carefully. 

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